FICHE DE TP 2 Exercice 1 restart; F := 3*X^5+7*X+6;

LCgqJEkiWEc2IiIiJiIiJEYkIiIoIiInIiIi

Delta := discrim(F,X);

IiolKSo+VVc=

ifactor(Delta);

KiwtSSFHNiI2IyIiIyIiJS1GJDYjIiIkRistRiQ2IyIjOCIiIi1GJDYjIiMkKUYvLUYkNiMiJGAqRi8=

f := F mod 5;

LCgqJEkiWEc2IiIiJiIiJEYkIiIjIiIiRik=

discrim(f,X) mod 5; # on utilise la sp\303\251cialisation \303\240 d\303\251faut de forme inerte du discriminant

IiIl

m:=degree(f,X);

IiIm

df:=diff(f,X) mod 5;

IiIj

k:=m-1-degree(df,X);

IiIl

R:=Resultant(f,df,X) mod 5;

IiIj

c:=lcoeff(f,X);

IiIk

(-1)^(m*(m-1)/2)*c^(k-1)*R mod 5;

IiIl

Exercice 2 restart; f := X^4 -4*X^3-X^2+ 16*X -12;

LCwqJClJIlhHNiIiIiUiIiJGKComRidGKClGJSIiJEYoISIiKiQpRiUiIiNGKEYsKiYiIztGKEYlRihGKCIjN0Ys

g := X^3 +2*X^2-X-2;

LCoqJClJIlhHNiIiIiQiIiJGKComIiIjRigpRiVGKkYoRihGJSEiIkYqRiw=

resultant(f,g,X);

IiIh

gcdex(f,g,X);

LCgqJClJIlhHNiIiIiMiIiJGKEYlRihGJyEiIg==

with(LinearAlgebra): S_:=SylvesterMatrix(f,g,X): S:=Transpose(S_);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtYWN0aW9uR0YkNiUtSShtZmVuY2VkR0YkNigtRiM2Ji1JJ210YWJsZUdGJDY7LUkkbXRyR0YkNiwtSSRtdGRHRiQ2KC1JI21uR0YkNiRRIjFGJ0Y5LyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Zgby8lK2dyb3VwYWxpZ25HRmBvLyUocm93c3BhbkdGXW8vJStjb2x1bW5zcGFuR0Zdby1GaG42KC1GW282JFEiMEYnRjlGXm9GYW9GY29GZW9GZ29GaW9GZ25GaW9GaW9GaW9GXm9GYW9GY28tRmVuNiwtRmhuNigtRiM2JS1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GaHAtRltvNiRRIjRGJ0Y5RjlGXm9GYW9GY29GZW9GZ29GZ25GaW8tRmhuNigtRltvNiRRIjJGJ0Y5Rl5vRmFvRmNvRmVvRmdvRmduRmlvRmlvRl5vRmFvRmNvLUZlbjYsLUZobjYoLUYjNiVGZHBGam5GOUZeb0Zhb0Zjb0Zlb0Znb0ZgcEZnbkZkcUZdcUZnbkZpb0Zeb0Zhb0Zjby1GZW42LC1GaG42KC1GW282JFEjMTZGJ0Y5Rl5vRmFvRmNvRmVvRmdvRmRxRmBwLUZobjYoLUYjNiVGZHBGX3FGOUZeb0Zhb0Zjb0Zlb0Znb0ZkcUZdcUZnbkZeb0Zhb0Zjby1GZW42LC1GaG42KC1GIzYlRmRwLUZbbzYkUSMxMkYnRjlGOUZeb0Zhb0Zjb0Zlb0Znb0ZqcUZkcUZpb0ZfckZkcUZdcUZeb0Zhb0Zjby1GZW42LEZpb0ZlckZqcUZpb0Zpb0ZfckZkcUZeb0Zhb0Zjby1GZW42LEZpb0Zpb0ZlckZpb0Zpb0Zpb0ZfckZeb0Zhb0Zjby8lJmFsaWduR1ElYXhpc0YnL0Zfb1EpYmFzZWxpbmVGJy9GYm9RJnJpZ2h0RicvRmRvUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGMS8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRl10LyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmh0LyUmZnJhbWVHRmh0LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGPS8lLWVxdWFsY29sdW1uc0dGPS8lLWRpc3BsYXlzdHlsZUdGPS8lJXNpZGVHRmZzLyUwbWlubGFiZWxzcGFjaW5nR0ZldC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGPUY5RjkvSSttc2VtYW50aWNzR0YkUSdNYXRyaXhGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zfdi8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRicvJSlydGFibGVpZEdRKzQ0ODg5MDc4NDJGJ0ZqdUZddkY5

H:=Transpose(ReducedRowEchelonForm(S_));

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

n,n:=Dimension(H);

NiQiIihGIw==

r:=Rank(H);

IiIm

[seq(add(Column(H,j)[i]*X^(n-i),i=1..n),j=1..r)];

NycsKCokKUkiWEc2IiIiJyIiIkYpKiYiI0BGKUYmRilGKSIjQSEiIiwoKiQpRiYiIiZGKUYpKiYiIzZGKUYmRilGLSIjNUYpLCgqJClGJiIiJUYpRikqJkYxRilGJkYpRilGKEYtLCgqJClGJiIiJEYpRikqJkY9RilGJkYpRi0iIiNGKSwoKiQpRiZGP0YpRilGJkYpRj9GLQ==

Exercice 3 restart; f := 3*X^5+7*X+6;

LCgqJEkiWEc2IiIiJiIiJEYkIiIoIiInIiIi

g := 2*X^5 -3*X^2+X-1;

LCoqJEkiWEc2IiIiJiIiIyokRiRGJyEiJEYkIiIiISIiRio=

resultant(f,g,X);

ISdyZTs=

P:=expand(quo(f*subs(X=Y,g)-subs(X=Y,f)*g,X-Y,X));

LEIqJkkiWEc2IiIiJUkiWUdGJSIiIyEiKiomRiRGJkYnIiIiISM2KiRGJEYmISM6KiZGJCIiJEYnRjBGKSomRiRGMEYnRihGLComRiRGMEYnRitGLiomRiRGKEYnRiZGKSomRiRGKEYnRjBGLComRiRGKEYnRihGLiomRiRGK0YnRisiI0BGJCIjPSomRiRGK0YnRiZGLComRiRGK0YnRjBGLiEjOEYrRidGOCokRidGJkYu

with(LinearAlgebra): n:=degree(f,X);

IiIm

B:=Matrix(n,n,(i,j)->coeff(coeff(P,X,n-i),Y,n-j));

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKS9nI0cm

Determinant(B);

ISdyZTs=

H:=Transpose(ReducedRowEchelonForm(B));

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKSc+MUQm

BezoutMatrix(f,g,X,method=symmetric);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKSFlMUQm

TTdSMApJNVJUQUJMRV9TQVZFLzUyODI2MDA0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjOiImIiYiIiFGJiEiKiEjNiEjOkYmRidGKEYpRiZGJ0YoRilGJkYmRihGKUYmIiNAIiM9RilGJkYmRishIzhGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzUyNTA2MTk2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjOiImIiYiIiIiIiFGJ0YnRidGJ0YmRidGJ0YnRidGJ0YmRidGJ0YnRidGJ0YmRidGJ0YnRidGJ0YmRiU=TTdSMApJNVJUQUJMRV9TQVZFLzUyNTA2NTgwWCwlKWFueXRoaW5nRzYjJSpzeW1tZXRyaWNHNiJbZ2whIiYhISEjMCImIiYiIiFGKCIiKkYpIiM2IiM6RipGK0YoISNARitGKEYoISM9IiM4Ric=